Integrand size = 30, antiderivative size = 314 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\frac {2 (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x) (d+e x)^{9/2}}-\frac {10 b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x) (d+e x)^{7/2}}+\frac {4 b^2 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)^{5/2}}-\frac {20 b^3 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^{3/2}}+\frac {10 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) \sqrt {d+e x}}+\frac {2 b^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)} \]
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Time = 0.07 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {660, 45} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\frac {4 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x) (d+e x)^{5/2}}-\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{7 e^6 (a+b x) (d+e x)^{7/2}}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{9 e^6 (a+b x) (d+e x)^{9/2}}+\frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x}}{e^6 (a+b x)}+\frac {10 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^6 (a+b x) \sqrt {d+e x}}-\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{3 e^6 (a+b x) (d+e x)^{3/2}} \]
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Rule 45
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5}{(d+e x)^{11/2}} \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^5 (b d-a e)^5}{e^5 (d+e x)^{11/2}}+\frac {5 b^6 (b d-a e)^4}{e^5 (d+e x)^{9/2}}-\frac {10 b^7 (b d-a e)^3}{e^5 (d+e x)^{7/2}}+\frac {10 b^8 (b d-a e)^2}{e^5 (d+e x)^{5/2}}-\frac {5 b^9 (b d-a e)}{e^5 (d+e x)^{3/2}}+\frac {b^{10}}{e^5 \sqrt {d+e x}}\right ) \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {2 (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x) (d+e x)^{9/2}}-\frac {10 b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x) (d+e x)^{7/2}}+\frac {4 b^2 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)^{5/2}}-\frac {20 b^3 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^{3/2}}+\frac {10 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) \sqrt {d+e x}}+\frac {2 b^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=-\frac {2 \sqrt {(a+b x)^2} \left (7 a^5 e^5+5 a^4 b e^4 (2 d+9 e x)+2 a^3 b^2 e^3 \left (8 d^2+36 d e x+63 e^2 x^2\right )+2 a^2 b^3 e^2 \left (16 d^3+72 d^2 e x+126 d e^2 x^2+105 e^3 x^3\right )+a b^4 e \left (128 d^4+576 d^3 e x+1008 d^2 e^2 x^2+840 d e^3 x^3+315 e^4 x^4\right )-b^5 \left (256 d^5+1152 d^4 e x+2016 d^3 e^2 x^2+1680 d^2 e^3 x^3+630 d e^4 x^4+63 e^5 x^5\right )\right )}{63 e^6 (a+b x) (d+e x)^{9/2}} \]
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Time = 2.27 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.90
method | result | size |
risch | \(\frac {2 b^{5} \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{e^{6} \left (b x +a \right )}-\frac {2 \left (315 b^{4} x^{4} e^{4}+210 x^{3} a \,b^{3} e^{4}+1050 x^{3} b^{4} d \,e^{3}+126 x^{2} a^{2} b^{2} e^{4}+378 x^{2} a \,b^{3} d \,e^{3}+1386 x^{2} b^{4} d^{2} e^{2}+45 x \,a^{3} b \,e^{4}+117 x \,a^{2} b^{2} d \,e^{3}+261 x a \,b^{3} d^{2} e^{2}+837 x \,b^{4} d^{3} e +7 e^{4} a^{4}+17 b \,e^{3} d \,a^{3}+33 b^{2} e^{2} d^{2} a^{2}+65 a \,b^{3} d^{3} e +193 b^{4} d^{4}\right ) \left (a e -b d \right ) \sqrt {\left (b x +a \right )^{2}}}{63 e^{6} \sqrt {e x +d}\, \left (e^{4} x^{4}+4 d \,e^{3} x^{3}+6 d^{2} e^{2} x^{2}+4 d^{3} e x +d^{4}\right ) \left (b x +a \right )}\) | \(282\) |
gosper | \(-\frac {2 \left (-63 x^{5} e^{5} b^{5}+315 x^{4} a \,b^{4} e^{5}-630 x^{4} b^{5} d \,e^{4}+210 x^{3} a^{2} b^{3} e^{5}+840 x^{3} a \,b^{4} d \,e^{4}-1680 x^{3} b^{5} d^{2} e^{3}+126 x^{2} a^{3} b^{2} e^{5}+252 x^{2} a^{2} b^{3} d \,e^{4}+1008 x^{2} a \,b^{4} d^{2} e^{3}-2016 x^{2} b^{5} d^{3} e^{2}+45 a^{4} b \,e^{5} x +72 a^{3} b^{2} d \,e^{4} x +144 x \,a^{2} b^{3} d^{2} e^{3}+576 x a \,b^{4} d^{3} e^{2}-1152 b^{5} d^{4} e x +7 a^{5} e^{5}+10 a^{4} b d \,e^{4}+16 a^{3} b^{2} d^{2} e^{3}+32 a^{2} b^{3} d^{3} e^{2}+128 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{63 \left (e x +d \right )^{\frac {9}{2}} e^{6} \left (b x +a \right )^{5}}\) | \(289\) |
default | \(-\frac {2 \left (-63 x^{5} e^{5} b^{5}+315 x^{4} a \,b^{4} e^{5}-630 x^{4} b^{5} d \,e^{4}+210 x^{3} a^{2} b^{3} e^{5}+840 x^{3} a \,b^{4} d \,e^{4}-1680 x^{3} b^{5} d^{2} e^{3}+126 x^{2} a^{3} b^{2} e^{5}+252 x^{2} a^{2} b^{3} d \,e^{4}+1008 x^{2} a \,b^{4} d^{2} e^{3}-2016 x^{2} b^{5} d^{3} e^{2}+45 a^{4} b \,e^{5} x +72 a^{3} b^{2} d \,e^{4} x +144 x \,a^{2} b^{3} d^{2} e^{3}+576 x a \,b^{4} d^{3} e^{2}-1152 b^{5} d^{4} e x +7 a^{5} e^{5}+10 a^{4} b d \,e^{4}+16 a^{3} b^{2} d^{2} e^{3}+32 a^{2} b^{3} d^{3} e^{2}+128 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{63 \left (e x +d \right )^{\frac {9}{2}} e^{6} \left (b x +a \right )^{5}}\) | \(289\) |
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Time = 0.29 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\frac {2 \, {\left (63 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 128 \, a b^{4} d^{4} e - 32 \, a^{2} b^{3} d^{3} e^{2} - 16 \, a^{3} b^{2} d^{2} e^{3} - 10 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} + 315 \, {\left (2 \, b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 210 \, {\left (8 \, b^{5} d^{2} e^{3} - 4 \, a b^{4} d e^{4} - a^{2} b^{3} e^{5}\right )} x^{3} + 126 \, {\left (16 \, b^{5} d^{3} e^{2} - 8 \, a b^{4} d^{2} e^{3} - 2 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 9 \, {\left (128 \, b^{5} d^{4} e - 64 \, a b^{4} d^{3} e^{2} - 16 \, a^{2} b^{3} d^{2} e^{3} - 8 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{63 \, {\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )}} \]
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Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\frac {2 \, {\left (63 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 128 \, a b^{4} d^{4} e - 32 \, a^{2} b^{3} d^{3} e^{2} - 16 \, a^{3} b^{2} d^{2} e^{3} - 10 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} + 315 \, {\left (2 \, b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 210 \, {\left (8 \, b^{5} d^{2} e^{3} - 4 \, a b^{4} d e^{4} - a^{2} b^{3} e^{5}\right )} x^{3} + 126 \, {\left (16 \, b^{5} d^{3} e^{2} - 8 \, a b^{4} d^{2} e^{3} - 2 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 9 \, {\left (128 \, b^{5} d^{4} e - 64 \, a b^{4} d^{3} e^{2} - 16 \, a^{2} b^{3} d^{2} e^{3} - 8 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x\right )}}{63 \, {\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )} \sqrt {e x + d}} \]
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Time = 0.30 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\frac {2 \, \sqrt {e x + d} b^{5} \mathrm {sgn}\left (b x + a\right )}{e^{6}} + \frac {2 \, {\left (315 \, {\left (e x + d\right )}^{4} b^{5} d \mathrm {sgn}\left (b x + a\right ) - 210 \, {\left (e x + d\right )}^{3} b^{5} d^{2} \mathrm {sgn}\left (b x + a\right ) + 126 \, {\left (e x + d\right )}^{2} b^{5} d^{3} \mathrm {sgn}\left (b x + a\right ) - 45 \, {\left (e x + d\right )} b^{5} d^{4} \mathrm {sgn}\left (b x + a\right ) + 7 \, b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 315 \, {\left (e x + d\right )}^{4} a b^{4} e \mathrm {sgn}\left (b x + a\right ) + 420 \, {\left (e x + d\right )}^{3} a b^{4} d e \mathrm {sgn}\left (b x + a\right ) - 378 \, {\left (e x + d\right )}^{2} a b^{4} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 180 \, {\left (e x + d\right )} a b^{4} d^{3} e \mathrm {sgn}\left (b x + a\right ) - 35 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) - 210 \, {\left (e x + d\right )}^{3} a^{2} b^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 378 \, {\left (e x + d\right )}^{2} a^{2} b^{3} d e^{2} \mathrm {sgn}\left (b x + a\right ) - 270 \, {\left (e x + d\right )} a^{2} b^{3} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 70 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 126 \, {\left (e x + d\right )}^{2} a^{3} b^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 180 \, {\left (e x + d\right )} a^{3} b^{2} d e^{3} \mathrm {sgn}\left (b x + a\right ) - 70 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 45 \, {\left (e x + d\right )} a^{4} b e^{4} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) - 7 \, a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )}}{63 \, {\left (e x + d\right )}^{\frac {9}{2}} e^{6}} \]
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Time = 10.63 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.32 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=-\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {14\,a^5\,e^5+20\,a^4\,b\,d\,e^4+32\,a^3\,b^2\,d^2\,e^3+64\,a^2\,b^3\,d^3\,e^2+256\,a\,b^4\,d^4\,e-512\,b^5\,d^5}{63\,b\,e^{10}}-\frac {2\,b^4\,x^5}{e^5}+\frac {10\,b^3\,x^4\,\left (a\,e-2\,b\,d\right )}{e^6}+\frac {x\,\left (90\,a^4\,b\,e^5+144\,a^3\,b^2\,d\,e^4+288\,a^2\,b^3\,d^2\,e^3+1152\,a\,b^4\,d^3\,e^2-2304\,b^5\,d^4\,e\right )}{63\,b\,e^{10}}+\frac {20\,b^2\,x^3\,\left (a^2\,e^2+4\,a\,b\,d\,e-8\,b^2\,d^2\right )}{3\,e^7}+\frac {4\,b\,x^2\,\left (a^3\,e^3+2\,a^2\,b\,d\,e^2+8\,a\,b^2\,d^2\,e-16\,b^3\,d^3\right )}{e^8}\right )}{x^5\,\sqrt {d+e\,x}+\frac {a\,d^4\,\sqrt {d+e\,x}}{b\,e^4}+\frac {x^4\,\left (63\,a\,e^{10}+252\,b\,d\,e^9\right )\,\sqrt {d+e\,x}}{63\,b\,e^{10}}+\frac {2\,d\,x^3\,\left (2\,a\,e+3\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^2}+\frac {d^3\,x\,\left (4\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^4}+\frac {2\,d^2\,x^2\,\left (3\,a\,e+2\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^3}} \]
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